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pmo problem 7

Source: PMO 2021

March 20, 2021
algebranumber theoryinequalitiesPhilippinesPMO

Problem Statement

Let a,b,c,a, b, c, and dd be real numbers such that a≥b≥c≥da \geq b \geq c \geq d and a+b+c+d=13a+b+c+d = 13 a2+b2+c2+d2=43.a^2+b^2+c^2+d^2=43.
Show that ab≥3+cdab \geq 3 + cd.
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